## Thursday, 14 March 2013

### The Away Goals Rule In Extra Time Is Fair To Both Teams.

Omar presented some very informative stats on his blog concerning the progress of teams in the knockout phases of such competitions as the Champions League, where away goals allow the visiting side to progress either at the end of normal and extra time. The only way that such ties can reach extra time is by the first leg score being identically reversed in the second match. In these instances, the home side has an extra 30 minutes of home advantage in which to progress and to balance this advantage, the away side can claim the decisive away goal if the goals scored in extra time are equally shared between the teams.

Omar's figures hint at the apparent fairness of this bargain because the number of home and away teams progressing without the need for a penalty shootout is reasonably equal. This is what we would expect if the quality of the home sides in the second leg was equivalent to that of the visitors over the course of the sample.

A theoretically based approach to the problem involves modelling the outcome of an extra 30 minutes between two equally matched sides to see if their chances of progressing in the extra time period is roughly equal. 1.4 goals per 90 minutes would be a typical goal expectancy for the home side compared to 1.0 for the visitors. However, over a 30 minute period these values would be greatly reduced. Figures of 0.57 goals and 0.41 would be typical values for the goal expectations of each team over final 30 minutes of the 90 in such a match up. So if both teams played in a similar manner to a normal game, these are the kind of goal expectancies we would see in extra time.

At the end of 90 minutes of the second leg, each team will have had an equal amount of playing time on their own turf to have built up a winning advantage. Therefore it is desirable that the extra thirty minute format shouldn't unduly favour one side over and above the difference in ability between the sides. So for two equally matched sides, the chances of progressing should be as near to 50/50 as possible.

Once extra time is reached, the home side has two routes to winning the tie. They can take and maintain a lead in extra time or they can keep the game scoreless and then progress on penalties. The away side has the same opportunities to reach the next phase, but also can progress with a score draw in the additional 30 minutes of play.

Chances Of An Equally Matched Home Side Progressing From Extra Time in the UCL.

 Team. Win in ET Draw ET period   0-0. Win Shootout. Overall Chance of Progressing. Home Team 32% 38% 50% 51%

Chances Of An Equally Matched Away Side Progressing From Extra Time in the UCL.

 Team. Win in ET Draw ET period 0-0. Score Draw in ET. Win Shootout. Overall Chance of Progressing. Away Team 21% 38% 9% 50% 49%

We can get a reasonable estimate of the chances of these individual outcomes occurring from a Poisson based calculation on the decayed pregame goal expectancy of our generic, equally talented home and away sides. For the shootout I've assumed each side has an equal chance of winning the penalty kick contest.

Either by accident or design, by allowing the away side the opportunity to still score an away goal in the additional 30 minute period, UEFA have almost entirely eliminated the home side's advantage of playing a larger proportion of the tie on home turf. The rules as they stand excellently perform the task of adding an extra half hour of potentially dramatic open play action, while still remaining fair to both sides.

1. Excellent post and I think your spot on when talking about fairness in extra time. Couple of things though, I think I've read before that there exists a home-advantage in penalty shoot-outs, so the probabilities there are not even (though I'm guessing it wouldn't change the overall chance of progressing by much). And I'm wondering whether it's fair to assume that the expected number of goals/minute in extra time is the same as in normal time given that players will be far more tired (and mistake prone) and that the away team may be more attack minded given the away goal rule. Would love to see if these affect the overall chance of progressing significantly.

2. hi 2nd.
Great points.
I agree the goal expectancy in ET is likely to be slightly different compared to a similar 30 minute chunk from normal play. Certainly more games go to a shootout in reality, indicating a degree of added caution from both teams.

Most penalty shootout studies I've seen don't statistically differ from a 50/50 proposition including the much quoted 60% advantage for a team shooting first. But I've certainly seen claims for both a presence and absence of HFA in shootouts.

Overall I think the conclusions are ball park accurate :-)

Once again, great points.

Mark.

3. Thanks for the reply Mark. Come to think of it, a penalty shoot-out could happen in front of the away end (thus negating some of the home advantage) and that is decided on a coin toss so 50/50 on penalties is probably fair.

Do you think away goals are fair across both legs in normal time? I know Jonathon Wilson has written recently to say he doesn't think they are, perhaps a move to away goals only counting in extra time would be a good solution?

4. Very interesting article! Intuitively, I always found the away goals rule during extra time somehow unbalanced, but you make a good point.

What I find unbalanced, however, is that UEFA counts the result of individual legs after 120min for the UEFA coefficients, and not after 90min. For example, a team that reaches extra time in the away leg after a home 0-1 defeat, but loses, say, 2-1 aET, gets 0 points in total and is not rewarded for the away win leading to extra time after all.

While I agree with you that the extra time after the 2nd leg is a fair procedure to determine the winner of the two-leg tie, it might not be fair to add it to the result of the 2nd leg alone to determine its outcome.

It might be a minor point, but do you have any opinion on that?

Stefan